By no means an expert, but there is an overlap with English Civil War sites, so I have visited all of these sites. These would probably be my 'highlights' of things to 'see'. Yes there are some significant sites that *should* be on the list e.g. Mortimer's Cross and Shrewsbury, but there isn't really anything to see there concerning the WotR. These suggestions will give you the most bang for your buck.
* York for the walls, Castle Museum, Yorkshire Museum, the Shambles, Cathedral & Towton. Pretty much everything in York is just brilliant. My favourite UK city, and always has been. There was a Richard 3 'experience' in one of the wall towers, but I have a feeling it shut recently.
* Leeds – Royal Armouries, and just down the road Wakefield and Sandal Castle (also a small exhibit of WotR archaeological finds in Wakefield Museum)
* Bosworth visitor centre, and the 'new' Bosworth Field site
* Leicester for the R3 visitor centre and the cathedral
* London – Tower of London, Wallace Collection (Wallace Collection was a private collection of, well pretty much everything, but particularly armour. wallacecollection.org )
* Tewkesbury, well presented battlefield
* Stoke Field, handily placed in Foundry Miniatures back garden
There is a gazeteer of WotR sites, apologies can't remember the author. Saw it on Caliver site when I was trawling their book lists (so try them and OMM first).
Don't forget the Battlefields Trust link for their battlefield hub, and WotR memorials database.
If you itinerary involves lots of castles and museums, it might work out considerably cheaper buying English Heritage, National Trust or Art Pass membership for a year. Generally speaking, if you are visiting 5 or more site cared for by one of the organisations (either English Heritage, or the National Trust) then you'll start saving money.
The Art Card is slightly more eclectic. Arts UK give grants to restore/save/buy art in conjunction with sites across the country. Those sites in return give free or discounted entry to Art Pass holders – this includes a number of museums, English Heritage and NT sites, galleries, stately homes in private ownership.
Once you have your itinerary it'll be time to work out if any of the memberships might be of use, saving you money. They might not.